March 2012 - mathfun@mailman.xmission.com

Re: [math-fun] big sunflowers
by Henry Baker 23 Mar '12

23 Mar '12

I don't know if "gimp" has 2D FFT capabilities, but there must be some open source package that does. Take some circles, hyperbolae, etc., compute, and look at the 2D FFT's. The graphics/image processing/lens design communities have done a huge amount of work w/2D FFT's. In fact, if you stand right up against a lens, you effectively see the 2D Fourier transform of the image. It's a little distorted, but it's there. At 12:00 PM 3/23/2012, James Propp wrote: >Most responses to my post have focussed on the frequency-domain, but what >about the specific patterns (circles, hyperbolas, etc.) that we see in the >spatial-domain? At some point these need to be explained. > >In fact, even leaving aside sunflowers, I'd like to see a nice mathematical >explanation of Moire patterns of the conventional kind. Has such a thing >ever been written? You'd think so, but I wasn't able to find one anywhere >on the web last time I checked! > >Jim Propp

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[math-fun] Digit echo forbidden -- last term?
by Eric Angelini 23 Mar '12

23 Mar '12

[if the lines are messy, please read this message here: http://www.cetteadressecomportecinquantesignes.com/DigitEchoForbidden.htm ] Hello Math-Fun, 1) If the n-th digit of S is "d" (d chosen from 0,1,2,3,...9) then the (n+d+1)th digit of S cannot be "d" 2) Start S with a(1)=1 3) Always extend S with the smallest available integer not yet present in S and not leading to a contradiction. forbidden "d" -------> 2 forbidden "d" ---> 0 2 1 2 1 1 1 2 1 5 2 forb "d" -> 1 2 3 4 5 16 07 18 9 2 03 3 22 1 4 13 2 15 42 1 62 6 7 1 2 87 1 39 S = 1,2,3,4,5,6,7,8,9,10,20,11,22,13,30,12,21,14,23,15,24,16,26,17,25,18,27,19,28,29,31,32,40,... S is obviously finite (the term 9876543210 blocks any extension, for example); what is the last term of S? Best, É.

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[math-fun] Documentary on MC Escher
by Stuart Anderson 23 Mar '12

23 Mar '12

1 hour youtube documentary on Escher; worth a look! http://www.youtube.com/watch?feature=related&hl=en&v=1d5blV9RDgM&gl=US Stuart Anderson

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Re: [math-fun] big sunflowers
by Henry Baker 22 Mar '12

22 Mar '12

In one dimension, the phenomenon is called "aliasing". Basically, the sampling theorem _requires_ that all frequencies above 1/2 the sampling frequency be removed _completely_ prior to sampling, else you get aliasing of these higher frequencies down to lower frequencies during the process of reconstruction. In particular, frequencies just above 1/2 the sampling frequency become very low frequencies on reconstruction. In 2D these low frequencies are called Moire patterns. The reconstruction filter also has to be perfect (allow no frequencies above 1/2 the sampling frequency), else it may induce artifacts, as well. In practise, it is essentially impossible to achieve perfect pre-sampling and reconstruction filters, so there will always be small amounts of aliasing in practise. Also, human eyes & ears seem to tolerate very small amounts of aliasing in order to get slightly sharper results in the higher frequencies (but still below the filter cutoff frequency). http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem At 11:24 PM 3/21/2012, James Propp wrote: >Is there a good place to read about the mathematics of MoirÃ© patterns? > >Jim > >On Wed, Mar 21, 2012 at 11:18 PM, Simon Plouffe <simon.plouffe(a)gmail.com>wrote: > >> Hello, >> >> Yes, this is the pixel problem, it is caused by the >> square grid of the image made of pixels. >> >> For example, when you draw lines that are close together >> on a pixel screen you get also MoirÃ© patterns, >> >> http://www.plouffe.fr/simon/**distributions%20modulo%201/** >> imagepages/image50.html<http://www.plouffe.fr/simon/distributions%20modulo%201/imagepages/image50.h…> >> >> Now the image of the sunflower and the MoirÃ© is a mixed effect >> between the natural occuring spirals and the 'closeness' of >> the points which causes the pattern, >> >> and of course : what is the formula or name of this >> pattern is a good question, >> >> Best regards, >> Simon Plouffe >> >> Le 22/03/2012 07:08, James Propp a Ã©crit : >> >>> Back in September my friend Joshua Burton sent me this email. He >>> and I and our mutual friend Michael Larsen exchanged a few emails >>> about the phenomenon, but we never followed up: >>> >>> http://www.cs.uml.edu/~jpropp/**sunflowers.html<http://www.cs.uml.edu/~jpropp/sunflowers.html> >>> >>> (I'm sending the URL because one of the imbedded images is quite large.) >>> >>> I asked Josh if it was okay to share this problem with others, and he >>> replied: >>> "By all means, share! The first to-do, I think, is to redo the work >>> independently of the Mma engine, to confirm that the moire patterns >>> are real, and not artifacts of some tool-specific rounding issue. If >>> it's real, I guess the next thing is to come up with some numerical >>> measure of the anomalous behavior of a big sunflower, as a function >>> of N. That, or an actual clue what's going on." >>> >>> Jim Propp

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[math-fun] Big multicolored sunflowers
by Warren Smith 22 Mar '12

22 Mar '12

Is there some sense in which the sunflower is a "quasicrystal"? E.g. what is its fourier transform? It plainly has no symmetry at all unlike, e.g. Penrose tiling with 5-fold symmetry. http://www.cs.uml.edu/~jpropp/sunflowers.html Another question is, suppose plant has not just "seeds" (points in plane it wishes to keep separated) but K different-colored seeds, and it likes to keep them separated and also to keep same-colored seeds further-separated. For example with K=2 the checkerboard would be a good solution, albeit it is not a sunflower. The sunflower with K=1 colors was based on the fact that the golden ratio g=(1+-sqrt(5))/2 is the "worst approximable" irrational number, i.e. min |P*g-Q| minimized over integer P,Q with |Q|<N, is large. It seems to me to get good sunflowers with K=2 colors, you want g to be replaced by h such that h only has poor approximations by P/Q if P is demanded to be even, which suggests considering -1+-sqrt(2) whose best rational approximations 1, 3/2, 7/5, 17/12, etc have always-odd numerators. One reason K=2 is interesting is we can now construct an "fake elliptic" function with poles at the red and zeros at the blue locations. If this were a checkerboard then that'd be a genuine elliptic (doubly periodic meromorphic) function, but with a bicolored sunflower we get a fake-elliptic function which is "quasiperiodic" (with whatever definition of "quasiperiodic" is valid for sunflowers). Certain "Painleve transcendent" functions have got zero & pole patterns that look somewhat like this, and it'd be great if we could make a connection. Warren D. Smith http://rangevoting.org

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Re: [math-fun] big sunflowers
by Henry Baker 22 Mar '12

22 Mar '12

I should have also added: With modern digital computer screens, the individual dots are "too" sharp, so you can actually see the square edges. Back in the days of CRT's (only available today in Chinese remainder bins) you could adjust the dots to overlap, to perform part of the task of the "reconstruction" filter. So with today's too-sharp computer screens, your own eyes form the last part of the "reconstruction" chain. The blurring of the dots can be performed by simply holding the screen far enough away so that the individual dots can no longer be distinguished. At 07:27 AM 3/22/2012, Henry Baker wrote: >In one dimension, the phenomenon is called "aliasing". Basically, the sampling theorem _requires_ that all frequencies above 1/2 the sampling frequency be removed _completely_ prior to sampling, else you get aliasing of these higher frequencies down to lower frequencies during the process of reconstruction. In particular, frequencies just above 1/2 the sampling frequency become very low frequencies on reconstruction. In 2D these low frequencies are called Moire patterns. > >The reconstruction filter also has to be perfect (allow no frequencies above 1/2 the sampling frequency), else it may induce artifacts, as well. > >In practise, it is essentially impossible to achieve perfect pre-sampling and reconstruction filters, so there will always be small amounts of aliasing in practise. Also, human eyes & ears seem to tolerate very small amounts of aliasing in order to get slightly sharper results in the higher frequencies (but still below the filter cutoff frequency). > >http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem > >At 11:24 PM 3/21/2012, James Propp wrote: >>Is there a good place to read about the mathematics of MoirÃ© patterns? >> >>Jim >> >>On Wed, Mar 21, 2012 at 11:18 PM, Simon Plouffe <simon.plouffe(a)gmail.com>wrote: >> >>> Hello, >>> >>> Yes, this is the pixel problem, it is caused by the >>> square grid of the image made of pixels. >>> >>> For example, when you draw lines that are close together >>> on a pixel screen you get also MoirÃ© patterns, >>> >>> http://www.plouffe.fr/simon/**distributions%20modulo%201/** >>> imagepages/image50.html<http://www.plouffe.fr/simon/distributions%20modulo%201/imagepages/image50.h…> >>> >>> Now the image of the sunflower and the MoirÃ© is a mixed effect >>> between the natural occuring spirals and the 'closeness' of >>> the points which causes the pattern, >>> >>> and of course : what is the formula or name of this >>> pattern is a good question, >>> >>> Best regards, >>> Simon Plouffe >>> >>> Le 22/03/2012 07:08, James Propp a Ã©crit : >>> >>>> Back in September my friend Joshua Burton sent me this email. He >>>> and I and our mutual friend Michael Larsen exchanged a few emails >>>> about the phenomenon, but we never followed up: >>>> >>>> http://www.cs.uml.edu/~jpropp/**sunflowers.html<http://www.cs.uml.edu/~jpropp/sunflowers.html> >>>> >>>> (I'm sending the URL because one of the imbedded images is quite large.) >>>> >>>> I asked Josh if it was okay to share this problem with others, and he >>>> replied: >>>> "By all means, share! The first to-do, I think, is to redo the work >>>> independently of the Mma engine, to confirm that the moire patterns >>>> are real, and not artifacts of some tool-specific rounding issue. If >>>> it's real, I guess the next thing is to come up with some numerical >>>> measure of the anomalous behavior of a big sunflower, as a function >>>> of N. That, or an actual clue what's going on." >>>> >>>> Jim Propp

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Re: [math-fun] big sunflowers
by Dan Asimov 22 Mar '12

22 Mar '12

I believe that indeed the arrangement of pixels on a computer screen does create a varying moire effect here. Moire patterns are closely analogous to interference patterns when waves are superimposed. (For one thing, when I magnify the many-dot pattern on my screen it's easy to see that the dots are quite irregularly shaped, rather than round disks.) BUT: I'm convinced that even if the rendition of the dots were mathematically ideal (i.e., no pixels involved), one would still see varying moire patterns here. Because almost everywhere, locally the dot pattern is well approximated by a linear lattice (i.e., the dots are at the corners of a tiling of the plane by identical parallelograms). * * * An interesting experiment that shows intriguing and mysterious moire patterns goes like this: Assume a computer screen with in {1..K} by {1..L} pixel coordinates denoted (x,y). Let the RGB color cube be [0,1]^3, and assign a discrete circular rainbow color table to Z/nZ = {0..n-1} for some large n; n = 1024 works well. Say by picking equally spaced points around the circle inscribed in the hexagonal intersection of the plane R+G+B = 3/2 with [0,1]^3.) Finally, color the pixel (x,y) the color whose number is x^2 + y^2 mod n. (I first tried this in 1981, and it's probably been done independently many times before and since.) --Dan << Back in September my friend Joshua Burton sent me this email. He and I and our mutual friend Michael Larsen exchanged a few emails about the phenomenon, but we never followed up: http://www.cs.uml.edu/~jpropp/sunflowers.html (I'm sending the URL because one of the imbedded images is quite large.) I asked Josh if it was okay to share this problem with others, and he replied: "By all means, share! The first to-do, I think, is to redo the work independently of the Mma engine, to confirm that the moire patterns are real, and not artifacts of some tool-specific rounding issue. If it's real, I guess the next thing is to come up with some numerical measure of the anomalous behavior of a big sunflower, as a function of N. That, or an actual clue what's going on." >> ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] Magic square expert?
by Dan Asimov 22 Mar '12

22 Mar '12

Rich wrote: << I have a note from a Korean high school student, asking if the results in his paper are new. The paper is about panmagic squares. Can anyone recommend an expert to consult? >> Je crois qu'il y a quelqu'un qui s'appelle Christian Boyer à < cboyer(a)club-internet.fr > qui maintiens le site web < multimagie.com >. Also someone named Harvey Heinz at < harveyheinz(a)shaw.ca > whose webpages at < http://www.magic-squares.net/ > discuss the subject. Finally there is also the Grogono Family Website with these pages: < http://www.grogono.com/magic/index.php >. --Dan ________________________________________________________________________________________ It goes without saying that .

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[math-fun] big sunflowers
by James Propp 22 Mar '12

22 Mar '12

Back in September my friend Joshua Burton sent me this email. He and I and our mutual friend Michael Larsen exchanged a few emails about the phenomenon, but we never followed up: http://www.cs.uml.edu/~jpropp/sunflowers.html (I'm sending the URL because one of the imbedded images is quite large.) I asked Josh if it was okay to share this problem with others, and he replied: "By all means, share! The first to-do, I think, is to redo the work independently of the Mma engine, to confirm that the moire patterns are real, and not artifacts of some tool-specific rounding issue. If it's real, I guess the next thing is to come up with some numerical measure of the anomalous behavior of a big sunflower, as a function of N. That, or an actual clue what's going on." Jim Propp

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[math-fun] 170 conjectural series for pi & other constants
by Warren Smith 22 Mar '12

22 Mar '12

http://arxiv.org/abs/1102.5649

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